The characteristic times of the transient stochastic dynamics with time-dependent control parameters distributed initial conditions

A systematic method is developed for the calculation of characteristic times, called nonlinear relaxation times (NLRT), to describe the dynamical relaxation of the linear transient stochastic systems whose control parameters are time-dependent functions. For those control parameters, which are modulated by a family of functions of the form a(t) = btδ −a0, with δ > 0, the method is applied to calculate the NLRT associated with the decay of unstable states of the linear stochastic systems when these parameters are continuously swept from below to above threshold (t- = (a0b)1δ). The ramp modulation is a model for which δ = 1, it is studied and formulated in terms of the time differences s = t − t_witht- = (a0b). The time scales of both models are compared under certain requirements of the involved parameters.